Apparatus previously designed for the exercise of the human body will typically resist the force applied by the exerciser in a linear manner. Note for example bar bell weight sets, or exercise equipment which incorporates weights and pulleys wherein the exerciser pulls on a rope and thereby slides a weight system upwardly and downwardly on a vertical rail. In these systems the resistance applied to the exerciser is constant regardless of the position of the exercise equipment relative to the exerciser's body.
In the last several years optimum exercise results have been obtained by the use of variable resistance exercise equipment. Such equipment applies a variable non-linear resistance to the exerciser during the motion associated with an exercise movement. Note for example Nautilus equipment that incorporates a variable radius cam located between the exerciser and the weights. Rotation of such a cam requires the application of increasing force through portions of the cam's rotation.
Most variable resistance exercise equipment is not easily transportable, however, due to its weight and complexity, and therefore does not satisfy the need of the general population for an easily transportable exercise apparatus. It would be improbable for example, for a person leaving on a business trip to easily transport an entire Nautilus equipment assembly.
A lightweight, and therefore apparently portable variable resistance exercise apparatus therefore needs to be developed. The design and development of such an apparatus should incorporate any available new technology. Due to the close proximity of such an apparatus to the human body, such an apparatus shoudld be safe to operate. Such an apparatus should also be easily manufacturable, lend itself to mass production, and have an acceptable longevity before failure.
In particular the longevity of the apparatus should allow enough cycles before failure to satisfy the purchaser's expectations for a piece of equipment that will last at least half a year or so. It can be easily calculated that such an exercise apparatus will be subjected to approximately 18,000 to 20,000 cycles during a six-month period of normal use. The cycles will vary in the strain imposed on the rod depending on the particular exercise being performed. The most severe strain is imposed when the rod is bent in a tear drop shape such that it's ends touch. Special consideration therefore need be directed to the selection of the material properties of such an exercise apparatus.
In related Application Ser. No. 030,397, the exercise apparatus comprised a flexible fiberglass rod formed from a mixture of a tough, hardenable resin system and essentially longitudinal fiberglass filaments. Gripping the rod at both ends and thereafter attempting to bend the rod until both ends touched one another required the application of increasing force. The variable resistance feature of the rod is caused by the increase in the strain imposed on the fiberglass filaments located on the outer periphery of the rod, as the radius of curvature of the rod is decreased.
To evaluate the possibility of meeting the fatigue life requirement of 18,000 cycles, rods of 1/4" diamter and 3/8" diamter were tested. A goal was to achieve about 10,000 cycles without failure where each cycle would see the rod bent to a tear drop shape with the ends touching. If this could be done then the consumer would be able to expect about 18,000 to 20,000 cycles where the strain of each cycle would vary from very small to the maximum which results when the ends of the rod are touched together. Various types of resins and fibers were used to fabricate the rods. The test results are given as follows:
______________________________________ 1/4Inch Diameter 5-Foot Long Rod Resin/Fiber Type Cycles to Failure Dow 411/E-glass 507 *IP8520/E-glass 1204 Ryton PPS/E-glass 2014 9310/S2-glass 6832 IP8520/PET fiber 60000 3/8Inch Diameter 5-Foot Long Rod Resin/Fiber Type Cycles to Failure *Dow 8084/E-glass 7 Dow 411/E-glass 8 Shell 828+871/E-glass 700 ______________________________________ *12% elongation resin
It is clear from the test results that the flexural fatigue performance of the circular cross section rods (both the 1/4 and 3/8-inch diameter rods) was substantially below the acceptable life of the exercise rods. In view of the fact that even the 1/4-inch diameter rods did not meet the fatigue life requirement, the plausibility of using fiber reinforced pultruded rods for exercise rods was, therefore, in doubt. Efforts were made to improve the fatigue life of the rods by using different types of resins including flexible high elongation resins. However, it is obvious from the test results that resin modification by itself would not be able to substantially improve the fatigue life of the rods to satisfy the 10,000 cycles life requirement. The use of different types of high performance fibers could enhance the fatigue life of the rods as demonstrated by the test results of the S2-glass rods. However, the cost of high performance fibers could jeopardize the marketability of the product. Although rods reinforced with polyester fiber did meet the fatigue life requirement of the exercise rods, one must bear in mind that polyester fiber by itself does not provide the appropriate stiffness performance of the rods and after repeated cycling the rod took a permanent bend (in the shape of an arc of a circle).
The maximum bending stress induced in the exercise rod bent into the shape of a teardrop, (FIG. 4), is given in Frisch-Fay, R., "Flexible Bars," London, Butterworths, 1962, pp. 1-11 as follows: EQU Maximum Bending Stress=2.19*.pi..sup.2 *E*(0.78c)/(4*L) (1)
where
2L=rod length PA1 E=longitudinal modulus of the rod PA1 2c=height of the rod cross-section
The glass content of the exercise rods was about 73-75% by weight or approximately 55-57% by volume. Hence, the following properties can be assumed for the undirectional composites in the exercise rods.
______________________________________ Rods with E-glass fiber E = 6.0 Msi X.sub.t = 1.40-1.55 ksi (ultimate tensile strength) Rods with S2-glass fiber E = 7.15 Msi X.sub.t = 230-243 ksi (ultimate tensile strength) ______________________________________
In general, the values of E and the ultimate tensile strength X will vary slightly with different resin systems. However, due to the lack of experimental data for the composite systems studied, they were assumed for the test to be the same for all resin systems.
Using Eqn. (1), the maximum flexural fatigue stresses induced in the 5-foot long 1/4-inch diameter E-glass and S2-glass rods are 105.4 ksi and 125.6 ksi, respectively. Although the values of the flexural fatigue stress for the E-glass and S2-glass rods are below the static tensile strength of the corresponding composites, they are too high to provide the required fatigue life of the exercise rods. To estimate the fatigue life of the rods at these fatigue stresses, it is necessary to have the fatigue curves of the various composites that were used in the exercise rods. In particular, it is more appropriate to have the fatigue curves of the composites made by the pultrusion process. It is obvious that such curves will not be readily available since it is expensive and time consuming to generate them. Hence, approximate relations between fatigue stresses and cycles to failure were used in this test for the E-glass and S2-glass composites, as set forth in Hahn, H. T., Hwang, D. G., and Chin, W. K., "Effects of Vacuum and Temperature on Mechanical Properties of S2-Glass/Epoxy," Recent Advances in Composites in the United States and Japan, edited by Vinson/Taya, ASTM STP 864, pp. 600-618. For the E-glass composites, we have EQU S.sub.F /X.sub.t =1.0-0.1* log.sub.10 (N) (2)
and for the S2-glass composites EQU S.sub.F /X.sub.t =1.115-0.154* log.sub.10 (N) (3)
where S is the fatique stress and N is the number of cycles to failure. It is appropriate to point out that Eqns. (2) and (3) are not arbitrary, but are based on known experimental data on some equivalent composite systems. A comparison between the predicted fatigue life using Eqns. (2) and (3) and the test results obtained are given below for both the 1/4 inch and the 3/8 inch diameter rods.
______________________________________ 1/4Inch Diameter 5-Foot Long Rod Actual Predicted Resin/Fiber Type Cycles to Failure Cycles to Failure Dow 411/E-glass 507 296-1585 *IP8520/E-glass 1204 296-1585 Ryton PPS/E-glass 2014 296-1585 9310/S2-glass 6832 4946-7655 IP8520/PET fiber 60000 -- 3/8Inch Diameter 5-Foot Long Rod Actual Predicted Resin/Fiber Type Cycles to Failure Cycles to Failure *Dow 8084/E-glass 7 1 Dow 411/E-glass 8 1 She11 828 + 871/E-glass 700 1 ______________________________________ *12% elongation resin
It can be seen from the comparison given above that reasonably good correlations can be obtained between the experimental results and the predicted life using Eqns. (1)-(3).
As can be seen, both the actual and predicted cycles to failure for both the 1/4-inch and 3/8-inch diameter rod are unacceptably low. An exercise apparatus therefore need be designed that has a maximum flexural fatique stress so as to insure an acceptable longevity, in the neighborhood of 18,000 cycles, to insure consumer acceptance of the durability of the apparatus, yet is stiff enough to provide a good workout for the exerciser.